We assume that the observable \(O\)  is known and the physical experiment \(\operatorname{{\mathcal {E}}}\)  is an unknown CPTP map that belongs to a (known) function class \(\operatorname{{\mathcal {F}}}\) . Apart from \(\operatorname{{\mathcal {E}}}\in \operatorname{{\mathcal {F}}}\) , the process can be arbitrary — a common assumption in statistical learning theory [1]}, [2]}, [3]}, [4]}, [5]}.
For the sake of concreteness, we assume that \(\operatorname{{\mathcal {E}}}\)  is a CPTP map from a Hilbert space of \(n\)  qubits to a Hilbert space of \(m\)  qubits.
Regarding inputs, we consider bit-strings of size \(n\) : \(x \in \left\lbrace 0,1\right\rbrace ^n\) . This is not a severe restriction, since floating-point representations of continuous parameters can always be truncated to a finite number of digits.
We now give precise definitions for classical and quantum ML; see Fig. REF  for an illustration.
